Next step is to make this setup work in the computer. You definitely don't want it to be powered by a molex connector, since they are only powered when computer is ON, and this is intended to turn it on when at power off state... That's when the WOL (wake on lan) connector is great.
WOL is intended to let the NIC tell the motherboard to power on when its address is called from the network. Not wanting to mess around with the signals on the motherboard, all you need is its 5V power that is always present when the motherboard is plugged.

WOL Cable

WOL Connector
The only thing we need to care is to get wires long enough to be able to place it into the computer, so that's when you should start worrying about the internal location of your remote start device. The WOL connector have 3 pins. 1-2-3 as for 5V, GND, Signal. we'll only use the 5V and ground, and they will be directly plugged replacing the batteries, just like we did with the molex connector. Same thing, different connector. I got my WOL wire from my NIC card. the green wire is signal and I don't use it. The black is ground, and the red 5V.

WOL Connecter Wiring
As for the output of the relay, the "switch" is intended to be placed in conjuncture with the power switch. Its to be plugged in parallel with it, so either the normal switch or the remote one could work. If they would be plugged in series, then both buttons would need being pressed at the same time. It could be a nice security measure, but since its a remote start we're building wiring them in parallel is better.

Relay and WOL Wiring Diagram
What I did is cut the power switch wires, solder the new connector from the relay (its a switch, any side is OK - its not polarized), plus the old casing's power switch altogether with the wires to the motherboard. (3-way solders. 2 wires going to A- motherboard, B- remote start's relay, C- power switch). If you don't want to use the usual power switch no more, you can just replace the old switch by the connectors from the relay.
Everything is now set, time to plug into your computer. You might want to make sure WOL is activated on your computer (usually a BIOS setting). When putting you install it into the computer, make sure you don't have any possible short circuits with the case, motherboard, or other devices. A small project box, some raiser bolts, static-free foam is suggested to insult it from your case. I personally bolted it 1/4 inch from the case, making sure nothing touches it.

Remote Start and Chime
And that's my remote start. This mod doesn't use the dremel, nor vinyl dye, but it was fun and challenging to build, and very useful for my mother. She never ever more uses the front bezel's button to power up the computer. I've been given the idea of modding the remote into the wireless keyboard, which could be really nice.
There is surely other ways of doing this, I always say that whatever you do on your computer is YOUR mod - keep it original, but I hope this was instructive for you as it was for me. If you have any trouble trying to make this work out, feel free to ask away on the forums.
Have fun, and keep customizing your stuff!